Sure it will not delete them automatically, but you can do it manually from the iPhone or another client. On the iPhone:
- go to mail app
- Go all the way to the top (left)
- At the bottom each account are listed, where you can access all folders each account may have
- Select the Cox account
- Click edit top right corner
- Click the folder (Apple call them mailboxes!?!) you want to delete
- Press delete at the bottom